Mathematical model of bone remodeling - Essay Example

? MATHEMATICAL MODEL OF BONE REMODELLING al affiliation 17th March Introduction For many years,bones have been known as complicated tissues which can be repaired and reconstructed in people’s lives continuously. In the last years, people have made it very clear that bone reconstruction relies on the relationship between two uncommon cells which are osteoclasts and osteoblasts at the cellular level. The osteoclasts refer to a variety of nuclear cells with hematopoietic origin. The second are mononuclear cells. The bone reconstruction process is made of two processes; the new bone’s formation and the old bone’s re-sorption. In the humans’ body, some auto-crine and paracrine parameters ruled the bone reconstruction process for the bone’s good. For the regulation of the cell types,according to the theory that whether the cell has the nuclear envelope, which is the boundary of the nucleus, the cell can be divided into prokaryotic and eukaryotic cells (Manolagas et al 1995, page 34). It’s noted that, the prokaryotic cells and eukaryotic can be separated like that. First, it’s the prokaryotic cells where the cells are smaller, no nuclear envelope, without nucleolus, without forming the nucleus; secondly, the eukaryotic cells where the cells are larger, and it have the nuclear envelope, nucleolus, a true nucleus; thirdly, the prokaryotes, which is comprised by prokaryotic cells constitute biological. Fourthly, the eukaryotes, which is from eukaryotic cells constitute biological. Bifurcations math’s theory According to Manologas (1995) bifurcations studies the characteristics of the bifurcation fact and its mechanism of the math theory. Bifurcation fact is a math fact which has all kinds of performance in nature. One parameter of the system is the continuity changing to a critical value, and the system’s global state.For instance, the qualitative properties of topological properties, will change suddenly for some full determination of the nonlinear system. This critical value is called the parameter, the bifurcation value or branch value. The reason why the researchers find this intriguing is that their cause is not in a bifurcation but in a change of the inactive-state concentrations in some bones with complications such as their not being firm. (Manolagas, 1995, pages 67). The systems theory has many strong tools to check and analyze the bifurcations. Researchers found a lot of math models for the regulation of bone reconstruction for the bifurcation properties in the present practice. These kinds of tools could be used for understanding the reasons for some kind of complications if it is found they have some relationship with it. Hence, the common models can give a broad prospect of the dynamics of the system, and it offers convenience to the specific models. More significantly, the analysis of the common model discovers dynamical instabilities that are very much related to pathologies of bone reconstruction. In summary, they used the common modeling ways and it permits analyzing models in which the kinetics is not constrained to the specific math functions. The assumption being put forward in the present time produces and extends to the 3D models are quiet if that assumption doesn’t exist; we are shown that it is in two-parameter models with the proposed structure, where the firmness of the inactive state demands that the actions of OPG control over those of RANKL.Those results show that osteoblast precursors have a significant effect on the dynamics and it ought to be taken into consideration in models. For human kind, the bone reconstruction process is influenced by some different parameters to maintain the bone balance. When the balance is broken, it may result into some different kinds of bone complications , for instance osteoporosis and Paget’s sickness of bone(Di Bernard, Feigin, Hogan & Homer,1999, page 1890). Paget’s sickness is common in the nipples of the elderly females and the surrounding skin, malignant tumors, which is also known as eczema carcinoma. Early treatment has an excellent prognosis, which can be found in men, and it’s often unilateral. It’s shown the infiltrating Perry class frigate erythematic and is scaly; it gradually expands to the surrounding skin. The slow development of early lesions and eczema are similar and they can be misdiagnosed. It is now clear that the breast eczema carcinoma consists mostly of breast intraductal carcinoma which develops in the epidermis. Consequently, all eczematoid carcinoma patients ought to undergo the spy breast palpation (Coleman, 2001). Osteoporosis occurs because of re-absorption cell dysfunction and the ceaseless hyperplasia hardens, which leads to bone nerve openings, a condition known as bones stenosis.It is caused by compression, particularly in the optic and acoustic nerves without stem cell transplantation in the life period of not more than six years. In addition, it’s an inherited sickness. The increasing of the bone’s density can severely cause the bone marrow cavity to be obliterated. Osteoporosis is a systemic disease which is characterized by the decline in bone mass and bone microstructure damage. The result is the increase of bone fragility, and fracture risk increases greatly, even though minor trauma cases are also prone to fracture. Osteoporosis is a chronic disease caused by multifactory. Before the fractures occur, it usually has no special clinical manifestation. The disease is popular in females than males and is common in postmenopausal women and the elderly (Di Bernard, Feigin, Hogan & Homer, 1999, page 1890). From the real parameter so near to bifurcations and from the math view, it means that the system ought to be strong against the parameter’s fluctuations, but it must be sensitive to the changes of the varieties. In a motive system, the strong answer of the inactive states to parameter change is often discovered near bifurcations, and it is a very important gate and critical point at which the firmness to perturbations is missing. Hence, it is very important that it ought to be many tradeoffs between dynamical firmness and response. Thus it is likely that the physiological state of the bone reconstruction system has the characteristics of some variable values near to a bifurcation point. Taking note of the bifurcation behavior of a 2-paremeter model topology, we showed that saddle-node bifurcations and Hopf bifurcations can happen here (Manolagas, 1995, pages 68). The nature wants the system to be sensitive so that external controllers, for example mechanical stress and many other parameters can make an impact on the reconstruction rate(May,1976).We can find out the parameters have the most significant impact on the system. In the model with two-variables, in order to describe the new cells’ maturation in a pool of precursors, we give a gain part (F, H, respectively) state parameter to, and the losing part (G, K)to describe the removing part of cells because of death and varieties further. By describing the population of osteoclasts, people might miss some significant information in a dynamic variable at that time osteoblasts and have different properties at maturation’s different period (Cheng, 1974, page 28). Because the power of precursor populations is denied, it can be debated that the two-variable models that put forward above is over simplified. In this model structure of literature, cells of osteoplastic lineage are shown by 2 dynamic parameter, responding osteoblasts (ROBs), R and active osteoblasts (AOBs),B. ROBs are forced to the osteoplastic lineage and acts with osteoclasts. And they are not still any osteoblasts. The model having a different structure has been put forward and was finally extended in. Some autocrine and paracrine parameters ruled the process of bone reconstruction in humans, which can maintain the bone’s balance itself. All of them play a significant function in the rule of bone reconstruction; people know that’s so necessary, and the RANKL is shown by cells of osteoplastic lineage, and it arrives at the RANKL. It is shown on the osteoplastic lineage cells especially when people found a signal way on the back catalyzer of NF-B (RANK) ;other cytokine osteoprotegerin (OPG), and its RANKL. Another significant adjustor, the cytokine TGF, is understood to affect osteoclasts and osteoblasts. And those processes are ruled by the decoy back, and it’s shown in osteoplastic cells and has the varieties of osteoclasts by constraining to RANKL and thus sequestering it. Over or under expression of TGF and the protagonists of the RANKL, the pathway has some related with some bone sick, for instance osteoporosis and Paget's sick of bone. The parameter like fc, hc and kc relies on the growing factor TGF, and when osteoclasts reabsorb the bone tissue, it is a significance adjustor in bone reconstruction, like TGF and so on. In the following, it makes the assumption that the all of the osteoblasts' life have not been influenced by some other adjustors. Hence, the decay part of osteoblasts is linear in b, and is independent of c. This changes into gc = 0 and gb = 1. Moreover, the researchers conclude there is no autocrine regulation of osteoblasts. It is released into the bone matrix, and it gives some convenience for the varieties of osteoblast progenitors to active osteoblasts, leading to fc > 0. Similarly, the decay part of osteoblasts is not influenced by osteoblasts, corresponding to kb = 0((Cheng, 1974, page 27)). Mathematical model construction A mathematical model to describe the bone reconstruction process ought to take into consideration the concentration of live osteoblasts and live osteoclasts; researchers begin that discussion of a mini animal model of 2 active parameters. Due to the fact that the smallest model may oversimplify the question by neglecting the number of precursors, particularly in the case of osteoblasts, an even complicated model for the role of osteoblasts is recovered at different steps of adult. Two variable models In the 2-parameter model, researchers specify 2 state parameters acquiring terms F and h for each to describe adults of new cells from a swimming pool precursor, and a wearing terms (g, K) are used to describe the removal of a cell because of death or further difference. So, it results to this fundamental equation Researchers say that, because that active precursor population is denied, this 2-parameter model put forward is oversimplified. Therefore, it would lose significant news, and it can describe a population of osteoclasts by only one active parameter, particularly at the time when osteoblasts have no common characteristics at different steps of being adults. A model with a different structure has been put forward hence being extended in. In this model, an osteoblast offspring is composed of 2 active parameters; response (ROB), osteoblasts and osteoblast activity (AOBS) B. ROBS nourish the osteoblast offspring and have an interaction with the osteoclast. However, there are no osteoblasts in the equation. There are 2 significant reasons to distinguish AOBS and ROBS: First, evidence from the experiment indicates osteoblasts can express RANKL and OPG in different steps of maturity, and those later steps of RANKL rate seem to reduce (Di Bernard, Feigin, Hogan & Homer, 1999, page 1890). Secondly, the TGF beta that exists on cells of the osteoblast offspring activates osteoblast differences only in those early steps of difference. But it seems to enlarge that pool for further inhibition of different osteogenic cells transforming the osteoblast into live osteoblasts. So that systemic property of bone reconstruction can be modeled as a system of ordinary differential equations (Cheng, 1974, page 29). Three variable models Researchers use a 3-dimensional model to express the problem. The schematic overview of that 3-parameter model is shown as follows FIG.2: skeleton overview of that 3-parameter model The structure of the 3-dimensional model translates to these equations Above it, 2 terms in each row corresponds to the acquiring and wearing of population in the respective cell types. Biological processes including the model of incentive equation, depends on these equations. Researchers interpret these processes in more detail in Jacobean after the formal establishment of it. Researchers can describe the structure model of that normalized equation as follows. Where, the lower-case parameters and equations care about the normalized variables and a1, a2, a3 are the characteristic timescales of responding osteoblasts (ROB), live osteoblasts (AOB), and osteoclast turnover. The equation for the 3-parameter model can be expressed as follows. This is the conclusion for the elasticity’s happening in the equation of Jacobean and the reference that researchers assign widely to them is expressed as follows. Researchers can use this mathematical ODE method to find the extreme and saddle point of the equation, these points are exactly what researchers want to compute and study (Di Bernard, Feigin, Hogan& Homer, 1999, page 1890). Conclusion and results The previous work has showcased the activities of the huge kind of models for bone reconstruction by applying the method of common modeling. Researchers know that both saddle-node bifurcations and Hopf bifurcations could happen. This model was used to complete the previous approach. According to that previous work, researchers have found that the most probable state when mature is close to bifurcations. The reason is that systems may be optimized for sensitivity to external parameters. People still don’t know whether the bifurcation phenomena connect with the known bone diseases. The analysis shown here suggests that Hopf and saddle-node bifurcations exist close to that physiological steady state. Researchers aren’t sure bone diseases would have any relationship with these bifurcations. The bifurcation happening in vivo should absolutely result to a pathological condition. So it appears to be a very reasonable relation, such that a Hopf bifurcation’s crossing and that of Paget’s onset may exist. It would imply that strong tools to study bifurcation theory and related data analysis could skillfully be used to look for that activity in sick people (Cheng, 1974, page 28). The elasticity’s SC, TC and WC can describe those nonlinearities in relation to the TGF. This way make the collection of AOBs both in the way that renders the difference of osteoblast progenitors to ROBs, which cause it to sc> 0 and in the method of forbidding that further difference of ROBs to AOBs stable, which cause it to tc< 0. In the researching model, researchers constrain sc to [0; 1] and tcto [1; 0]. The wide contains the option of Hill equations with the exponents as 1 and earlier models used it. The autocrine regulation’s essence of osteoclasts has not yet been finally understood. So researchers constrain wc to [0:5; 1:5](Di Bernard, Feigin, Hogan & Homer, 1999, page 1890). This wide is in that center of wc = 1, for if extra feedback is lack, a straight decrease term would happen. Especially if that additional feedback is negative (positive), that parameter is smaller (greater) than other. Researchers notice that the equation formed is assumed, which leads to strong straight decay (wc>(Di Bernard, Feigin, Hogan & Homer, 1999, page 1890). The law of osteoclasts in osteoplastic offspring’s cells is middle way putting into consideration the way of RANKL/RANK/OPG. It relies on that rate in the range of RANKL and its decoy receiver OPG, which related elasticity ’svr and vb as over zero or not. It contains a mixture of RANKL and OPG of related osteoblasts or live osteoblasts. Especially the literature described 2 fundamentally variable methods, and such as it discussed as models M1 and M2. These methods have characteristics of the general model, and these are as follows: When it satisfies vr< 0; vb> 0. Osteoblasts have relation with OPG, and that RANKL is shown by live osteoblasts. When it satisfies vr> 0; vb< 0. Osteoblasts have relation with RANKL, and live osteoblasts show the OPG. That middle conditions owning a differential expression of OPG and RANKL, and that have no that assumption that that exclusive expression can be found in our writing such as vr>vb> 0. In the above section, researchers show the conclusion of the model, and introduce them in the preceding section. The Bifurcation of the 2 modes can be calculated from that Jacobian. In 2-parameter model of bifurcation analysis, the result depends on 3 variables, and it can be visualized and understood in intuitive. But in that 3-parameter model, the larger number of parameters to obtain an intuitive is difficult to understand. Therefore, the researchers’ depend on the preliminary exploration of that common model when it comes to numerical procedures, which provides results that are easier to explain. For the impact of the model, it can be concluded as follows.The reproductive number offers a method for evaluating an infectious sick person, whether he can go on and persist in a given host population or not. Similar methods could be used for the emergence of new kinds in a mature group. So the researchers puts up with a method in case of influence on arriving new kinds or pathogens on the mature system. Moreover, they consider the difference in the steady abundances of local kinds at the time that the system faces a small fixed abundance or density of the new kinds. Pinning the new kind density to a specific value makes up with a press perturbation of the system (Di Bernard, Feigin, Hogan & Homer, 1999, page 1890). For calculating the impact, the researchers put forward formalism similar to the theory of metabolic control from the systems biology. The researchers show the impact of new kinds in widely-defined kinds of systems, which can be seen as a function of parameters that has a clear ecological interpretation by applying the thoughts of common modeling. They put forward formalism, which can be used in many-parameter models, for instance the complicated food webs or models for different stages in the development process. Reference list: Manolagas, R. J(1995), “Bone marrow, cytokines, and bone reconstruction”, The New England.Journalof medicine, vol.5, No.6 .pages.67-90. Cheng, H.(1974), “Origin, varieties and renewal of the four main epithelial cell types in the Mouse small intestine V. Unitarian theory of the origin of the four epithelial cell types”, American Journal of Anatomy, vol.1, No.2, pp. 27-50. Di Bernardo M.; Feigin M.I.; Hogan S.J. & Homer M.E (1999), “Local Analysis of C- Bifurcations in n-Dimensional Piecewise”, Journal of Smooth Dynamical Systems, Chaos, Solitons and Fractals, Vol.10, No 11, pp. 1881-1908. Read More